STA2023 Review:
Point Estimation

Introduction: Topics

• Basic descriptives:
  • Continuous variables:
    • Mean
    • Median
    • Variance and standard deviation
    • Range and interquartile range
  • Categorical variables:
    • Count
    • Overall percentage
    • Row percentage
    • Column percentage

Introduction: Data

• We will be using data from the kingdom of Equestria (yes, from My Little Pony).

• Mane Six:

  • Twilight Sparkle (Unicorn \to Alicorn)
  • Applejack (Earth Pony)
  • Fluttershy (Pegasus)
  • Pinkie Pie (Earth Pony)
  • Rainbow Dash (Pegasus)
  • Rarity (Unicorn)

Introduction: Data

Introduction: Data

• Name: the pony’s name
• Type: type of pony (Earth, Pegasus, Unicorn, Alicorn)
• Sex: sex/age of pony (Coal, Filly, Stallion, Mare)
• Flying speed: average flying speed (km/hr) for winged ponies
• Friendship: a harmony index from friendship activities (0-10)
• Magical energy: measured magical energy output (sparkles) for magical ponies
• Tail shimmer: how much light reflected by the pony’s tail (lux)

Types of Variables: Qualitative

• A qualitative or categorical variable classifies an observation into one of two or more groups or categories.
  • Nominal: purely qualitative and unordered
  • Ordinal: data can be ranked, but intervals between ranks may not be equivalent
• Examples:
  • satisfaction rating
  • favorite color
  • type of pet
  • education level
  • blood type

Types of Variables: Quantitative

• A quantitative or continuous variable takes numerical values for which arithmetic operations such as adding and averaging make sense; typically has a unit of measure.
  • Interval: meaningful differences between values, but no true zero point
  • Ratio: meaningful differences and a true zero point
• Examples:
  • age (years)
  • temperature (Celsius)
  • daily hours of sleep
  • ACT or SAT score
  • height (inches)

Types of Variables: Example

• Name: the pony’s name
• Type: type of pony (Earth, Pegasus, Unicorn, Alicorn)
• Sex: sex/age of pony (Coal, Filly, Stallion, Mare)
• Flying speed: average flying speed (km/hr) for winged ponies
• Friendship: a harmony index from friendship activities (0-10)
• Magical energy: measured magical energy output (sparkles) for magical ponies
• Tail shimmer: how much light reflected by the pony’s tail (lux)

Describing Data: Why?

• Why do we describe data? We want to tell a story!
  • Summarize n observations into a single description
  • Understand what is in the data
  • Spot patterns, missing data, or outliers
  • Compare groups or spot differences or oddities

Describing Data: How?

• How do we describe data?
  • Numbers
    • Frequency table
    • Mean & standard deviation
    • Median & IQR
  • Graphs
    • Bar charts
    • Box plots
    • Histograms

Point Estimation: Mean

• Mean: the average of a set of values

\bar{y} = \frac{\sum_{i=1}^n y_i}{n}

• Find the mean for the flying speeds (km/hr) of 5 ponies: {10, 20, 30, 40, 100}

\bar{y} = \frac{\sum_{i=1}^n y_i}{n} = \frac{10 + 20 + 30 + 40 + 100}{5} = 40

• The average flying speed for winged ponies is 40 km/hr.

Point Estimation: Median

• Median: The middle value in an ordered dataset.
  • When we have an even number of observations, we average the two middle.
• Find the median for the flying speeds (km/hr) of 5 ponies: {10, 20, 30, 40, 100}
  • First, we sort the data: {10, 20, 30, 40, 100}
  • Then, find the middle number: 30
• The median flying speed for winged ponies is 30 km/hr.

Point Estimation: Variance

• Variance: A measure of spread; the average of squared differences from the mean.
  • Higher variance = data has more spread.
  • In squared units of the data.

s_y^2 = \frac{\sum_iy_i^2 - (\sum_iy_i)^2/n}{n-1}

• Find the variance for the flying speeds (km/hr) of 5 ponies: {10, 20, 30, 40, 100}

s_y^2 = \frac{\sum_iy_i^2 - (\sum_iy_i)^2/n}{n-1} = \frac{(10^2+...+100^2)-(10+...+100)^2/5}{4} = 1250

• The variance is 1250 (km/hr)²

Point Estimation: Standard Deviation

• Standard Deviation: A measure of spread; the average distance from the mean.
  • Higher standard deviation = data has more spread.
  • Same units as the data.

s_y = \sqrt{s_y^2}

• Find the standard deviation for the flying speeds (km/hr) of 5 ponies: {10, 20, 30, 40, 100}

s_y = \sqrt{s^2_y} = \sqrt{1250} \approx 35.36

• The standard deviation is 35.36 km/hr.

Point Estimation: Range

• Range: difference between the maximum and minimum values

\text{range} = \text{max}(y) - \text{min}(y)

• Find the range for the flying speeds (km/hr) of 5 ponies: {10, 20, 30, 40, 100}

\begin{align*} \text{range} = \text{max}(y) - \text{min}(y) = 100 - 10 = 90 \end{align*}

• The range of the flying speeds is 90 km/hr.

Point Estimation: Interquartile Range

• Interquartile Range (IQR): range of the middle 50% of the data.

\text{IQR} = \text{P}_{75} − \text{P}_{25}

• Find the IQR for the flying speeds (km/hr) of 5 ponies: {10, 20, 30, 40, 100}
  • Recall that the median is 30.
  • We then find P_{25} using {10, 20} and P_{75} using {40, 100}
  • Thus, P_{25} = 15 and P_{75} = 70.

\begin{align*} \text{IQR} = \text{P}_{75} − \text{P}_{25} = 70 - 15 = 55 \end{align*}

• The IQR of the flying speeds is 55 km/hr.

Point Estimation: Proportion

• Proportion: a type of mean for categorical data
  • Often expressed as a percentage
    
  • Useful for categorical responses

\hat{p} = \frac{\sum_{i=1}^n y_i}{n},

• Note that in this case,

y_i = \begin{cases} 1 & \text{if in category }i \\ 0 & \text{otherwise} \end{cases}

Point Estimation: Proportion

• Find the proportion of ponies that have wings in the following sample: {Y, N, Y, Y, N, Y}

• Count the number of “Y” responses and divide by total:

\hat{p} = \frac{\sum_{i=1}^n y_i}{n} = \frac{4}{6} \approx 0.67

• The proportion of ponies with wings is 0.667 (or 66.7%).

Point Estimation: Frequency Table

• Frequency table: A table showing how often each value appears in a dataset.
  • Useful for categorical responses.
  • For each category, i, we report n_i (\%_i)
• Find the freqency table for the following sample of 8 ponies: {Earth, Pegasus, Unicorn, Earth, Pegasus, Pegasus, Unicorn, Alicorn}

Point Estimation: Frequency Table

• Putting this into a table,

Point Estimation: Contingency Table

• Contingency table: A table that summarizes two qualitative variables and their overlap.

• We will not concern ourselves with the derivation, but will rely on R.

• Consider this data,

Point Estimation: Contingency Table

• The resulting contingency table would look someting like this:
  • We are using column totals as our denominators.

# A tibble: 4 × 3
  pony_type No        Yes      
  <chr>     <chr>     <chr>    
1 Alicorn   0 (0.0%)  1 (25.0%)
2 Earth     2 (50.0%) 0 (0.0%) 
3 Pegasus   0 (0.0%)  3 (75.0%)
4 Unicorn   2 (50.0%) 0 (0.0%) 

Graphs: Box Plots

• Box plots display the distribution of a continuous variable using the five number summary:

  • Whisker: Minimum
  • Beginning of box: 25^th percentile (first quartile; Q1, P25)
  • “Middle” of box: Median (50th percentile, second quartile; Q2, P50)
  • End of box: 75^th percentile (third quartile; Q3, P75)
  • Whisker: Maximum

• We use box and whisker plots to get an idea of the spread and skewness of the data.

• Note: there are different ways to define the whiskers.

  • I use the min/max as whiskers when sketching by hand.
  • ggplot() uses 1.75 \times IQR.

Graphs: Box Plots

• Describe this box plot:

Graphs: Box Plots

• Describe this box plot:

Graphs: Box Plots

• Describe this box plot:

Graphs: Box Plots

• Describe this box plot:

Graphs: Histograms

• Histograms show the distribution of a continuous variable.

  • What is the shape of the distribution?
  • Is the distribution symmetric? Skewed? How skewed?

• Values are grouped into intervals (“bins”), then the bin height demonstrates how many values fall into that interval.

• This allows us to quickly see if there are any oddities.

  • Increased proportion of a specific value/bin.
    • Zero inflation? Value used to indicate missing?
  • Any values that are “out in the tail”.
    • Outlier? Data entry error?

Graphs: Histograms

• Describe the histogram:

Graphs: Histograms

• Describe the histogram:

Graphs: Histograms

• Describe the histogram:

Graphs: Histograms

• Describe the histogram:

Graphs: Histograms

• Describe the histogram:

Graphs: Bar Graphs

• Bar graphs display the distribution of categorical data.
  • The frequency or proportion of observations is displayed on the bar graph.
• Bar graphs usually have categories on the x-axis and counts or proportions on the y-axis.
  • Note that we could flip the axes to create a vertical bar graph.
• Note that the bars are separated on the x-axis to indicate the lack of continuity.

Graphs: Bar Graphs

• Consider the bar graph, below.

Graphs: Side-by-Side Bar Graphs

• Consider the bar graph, below.

Graphs: Stacked Bar Graphs

• Consider the bar graph, below.

Graphs: Histograms vs Bar Graphs

• We have now reviewed two “bar style” graphs that we see regularly: histograms and bar graphs.

• We use histograms to see the distribution of continuous variables.

  • The x-axis represents numeric intervals.
  • The bars touch each other to represent continuity.

• We use bar graphs to see the distribution of categorical variables.

  • The x-axis represents categories.
  • The bars do not touch each other, implying distinct categories.

Graphs: Scatterplots

• Scatterplots allow us to look at the relationship between two continuous variables.
  • Each point on the graph represents one observation.
• What statisticians use scatterplots for:
  • Explore patterns (aka trends or relationships).
    • Linear relationships.
    • Non-linear relationships.
  • Detect clusters of observations.
  • Find oddities in the data (outliers).
• When we describe the relationship, we are really answering the question, “As x increases, what happens to y?”

Graphs: Scatterplots

• Consider the scatterplot, below.

Graphs: Scatterplots

• Consider the scatterplot, below.

Graphs: Scatterplots

• Consider the scatterplot, below.

Graphs: Scatterplots

• Consider the scatterplot, below.

Graphs: Scatterplots

• Consider the scatterplot, below.

Wrap Up

• We have covered (“reminded” ourselves of) a lot today!
  • Always remember that I do not expect you to:
    • Memorize code.
    • Produce code in a timed environment.
    • Automatically know how to do these things.
  • I do expect you to:
    • Use your resources (lecture slides, GitHub website, Discord).
    • Try your best.

Wrap Up

• Today’s lecture:
  • Basic summarization of data.
  • Basic data visualization.
• This week’s lab:
  • Summarizing data
  • Visualizing data
• Next week:
  • Review of statistical inference.
  • Confidence intervals and hypothesis tests.
    • One sample means.
    • Two sample means.
      • Independent data.
      • Dependent data.

Wrap Up

• Daily activity: the .qmd we worked on during class.
  • Due date: Monday, June 23, 2025.
• You will upload the resulting .html file on Canvas.
  • Please refer to the help guide on the Biostat website if you need help with submission.